\section*{Boundary Element Method for Elasticity}

In elasticity theory, there are two spaces, namely material space \textbf{X} and world space \textbf{x}. Material space is like the rest configuration space for object, which is undeformed. World space is like the space for object under deformation. A mapping called displacement mapping $\textbf{u}: \textbf{X} \mapsto \textbf{x}$ connects these two spaces. To measure the strain, we need something called deformation gradient $\textbf{F} = \nabla \textbf{u}$. Two popular strain people use are Green strain $\boldsymbol{\varepsilon} = \textbf{F}^T\textbf{F}$, which is nonlinear but invariant under rigid body motion, and Cauchy strain $\boldsymbol{\varepsilon} = \frac{1}{2}(\textbf{F}^T + \textbf{F})$, which is linear and approximately correct under small deformation.

To map from strain to stress, there are lots of different material model describing different material property. One of the most popular and easy linear material model is by using Young's modulus and Poisson ratio, which is also used in Pai's paper. These two parameter control the rigidity and compressibility respectively and people can form a fourth order tensor, say $\mathbb{C}$, based on them. Then we can compute the stress $\boldsymbol{\sigma} = \mathbb{C} : \boldsymbol{\varepsilon}$. Here stress is a second order tensor which encodes the internal force flux. One can compute the force at a point inside the material in a certain direction $\textbf{p}$ by $\boldsymbol{\sigma}\cdot\textbf{p}$. To compute the net force at a point, one can use $\nabla\cdot\boldsymbol{\sigma}$.

Thus it becomes clear to us that, basically what the Navier's equation, so called in Pai's paper, tells us is the following

\begin{equation}\nonumber
\nabla\cdot\boldsymbol{\sigma} = \nabla\cdot(\mathbb{C}:\boldsymbol{\varepsilon}) = \textbf{f}_{int} = \textbf{f}_{ext} = \textbf{b}
\end{equation}

If we forget the constant fourth order tensor $\mathbb{C}$ for a while, we have

\begin{equation}\nonumber
\nabla\cdot\boldsymbol{\sigma} = \nabla\cdot\boldsymbol{\varepsilon} = \frac{1}{2}\nabla\cdot(\textbf{F}^T + \textbf{F}) = \frac{1}{2}\nabla\cdot(\nabla\textbf{u}^T + \nabla\textbf{u})
\end{equation}

If you expand the above equation, you will get what is described in Navier's equation igonoring the coefficients which can be added back by taking $\mathbb{C}$ into account.